# Finding Determinant By Cross Multiplication

In the previous article, we have seen how Determinant decides whether a system of equation (read square matrix) has inverse, or it has a solution, only when the determinant is not zero. The determinant is obtained from the equation given below.

$determinant = \sum \pm a_{1\alpha}a_{2\beta}...a_{nv}$

To know more about finding determinant in this way , read previous article. Here we will discuss about finding determinant by cross multiplication but before that let us understand the different notations used to represent determinants.

### Notation For Determinants

There are several notation for determinants given by earlier mathematicians. Suppose $A$ represents a augmented matrix from a system of linear equations, then determinant of $A$ is given below.

Let the matrix $A$ be a 2 x 2 matrix.

$A = \begin{bmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{bmatrix}$

Different ways to represent determinant of matrix $A$$det(A)$  -- (1)

$|A|$ -- (2)

$\begin{vmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{vmatrix}$ -- (3)

### Determinant as a Function

Imagine determinant to a function that take a square matrix as input and give a single value as output. For example,$f(x) = x^3$ be a function where $x$ could be any real number. Similarly, $det(A)$ is a function that matrix as input and give a determinant value $d$. The determinant value is always integer because it is linear combination of integers, that is, all values are integers in the matrix.

Determinant of $1 \times 1$ Matrix

If $A$ is a matrix with just one element, then its determinant is the same element.

Example #1

Let $A$ be a square matrix of order $1 \times 1$$A = \begin{bmatrix}2\end{bmatrix}$

Then the determinant of $A$ is

$|A| = |2| = 2$

### Determinant of $2 \times 2$ Matrix

The determinant of a $2 \times 2$ matrix is obtained by performing cross multiplication. See the following figure.

Example #2

Let $A$ be a $2 \times 2$ square matrix. Find the determinant of the matrix $A$.

Solution:

Let the $A$ be 2 x 2 square matrix.

$A = \begin{bmatrix}2 & 3\\1 & 5\end{bmatrix}$$|A| = a \times d - b \times c$$|A| = 2 \times 5 - 3 \times 1$$|A| = 10 - 3 = 7$

Example #3

Let $B$ be a square matrix of order $2 \times 2$. Find the determinant of the matrix $B$.

Solution:

Let $B$ be a square matrix of order $2 \times 2$.

$B = \begin{bmatrix}5 & -1\\4 & -3\end{bmatrix}$$|B| = a \times d - b \times c$$|B| = 5 \times (-3) - (-1) \times 4$$|B| = (-15) - (-4)$$|B| = (-15) + 4 = -11$

### Determinant Of $3 \times 3$ Matrix

The determinant of a $3 \times 3$ matrix is also possible through cross multiplication; Since we have a larger matrix we need to convert the larger matrix into smaller matrix to compute determinant. See figure below.

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Finding Determinant By Cross Multiplication

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